The Mathophile

         There was a big empire of king Ramulo somewhere in the middle of Asia. He was the most powerful king in whole continent. Also, he was wealthiest king of all the kings and people in his country where happy and everyone was rich enough. Nobody was poor and beggar there. And obviously the king also proud of this. But, eventually he was becoming arrogant and egoistic on his wealth and Empire as a king. Likewise the people in his Empire where following the same.           The king Ramola had a daughter named Jasmine . She was very charming and beautiful. One day, Jasmine was wondering in the small village nearby. Suddenly she was caught by some soldiers of king's enemy's army. The brave and intelligent man named Akira saved Jasmine from enemies using his brain and took her to the king. king was worried about his daughter. As he saw her daughter he became very happy. He was impressed by Akira . He appealed Akira to ask for anyt...

    Well, we all know that the sum of first n natural numbers is  n(n+1)/2.  Fine! But the mystery is, what is the sum of all counting numbers? Well, this is no longer a mystery, as the great Indian mathematician Srinivasa Ramanujan gave the sum for the same.                                Srinivasa Ramanujan          He suggested that the sum of all counting numbers (all natural numbers upto ♾️) i.e.  1+2+3+4+5+6+7+ ••• upto ♾️ is  -1/12.  If you still feel it difficult to believe, if you think, how all positive numbers sum up to negative number? Well, don't think too much, keep reading my article to unreveal all the mysteries.          The following two series are the backbone of our proof for the sum: 1-1+1-1+1-1+1-1+1-1+ •••• = 1/2 1-2+3-4+5-6+7-8+9-10+ ••• = 1/4     ...

Grand's Series: Look at the following series, S = 1  − 1 + 1 − 1 + ⋯  which can be also represented as  is called as  Grandi's Series . The question that came to the ancient mathematician was,  what is the sum of the series?   Well, these mathematician came up with the two approaches as follows: 1 . They attack on the series like,   (1 − 1) + (1 − 1) + (1 − 1) + ... = 0 + 0 + 0 + ... = 0. 2 . On the other hand some treat it like,  1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + ... = 1 + 0 + 0 + 0 + ... = 1. Well, both the results were very contradictory. By treating the above series in two different ways, one can get either 0 or 1 as an answer. In 1703, one Italian mathematician named  Guido Grandi  came up with the idea that the sum of above  series is neither 0 nor 1,  but it is  1/2 . After which many mathematician of the time disagreed and said, "what? The sum can't be 1/2." This was very hot topic for the debate in th...